论文标题
非共性ornstein-uhlenbeck的参数估计
Parameter estimation of non-ergodic Ornstein-Uhlenbeck
论文作者
论文摘要
在本文中，我们考虑了由General Gaussian流程$（g_t）_ {t \ ge 0} $驱动的非共性ornstein-uhlenbeck〜（o-u）过程的漂移参数$θ$的统计推断。当$ h \ in（0，\ frac 12）\ cup（\ frac 12,1）$ $ r（t，s）= e [g_t g_s] $的第二阶混合部分衍生物可以分解为两个部分，其中一个与$ brownian Motion（FBM）以及$ $ $ $ | the $ | | tss c | the | the | |这种情况涵盖了大量常见的高斯过程，例如FBM，次骨后布朗运动和双分裂的布朗运动。在这种情况下，我们验证$（g_t）_ {t \ ge 0} $满足参考文献中的四个假设\ cite {el2016}，也就是说，噪声具有Hölder连续路径；噪声的差异是由功率函数界定的。在Ergodic O-U进程$ X $的情况下，解决方案$ X_T $的渐近方差存在，严格为$ t \ to \ infty $；对于固定的$ s \在[0，t）$中，噪声$ g_s $在$ t \ to \ infty $上渐近地独立于ergodic解决方案$ x_t $，因此，基于连续观察$ x $，请确保估算值$ \tildeθ_t$的强差和渐近分布。验证$（g_t）_ {t \ ge 0} $满足参考\ cite \ cite {es-sebaiy2019}中的假设，即增量过程的差异$ \ \ {ζ_{ζ_{ζ_{t_i} -t_i} -m {t_ {t_t_ {t_t_ {t_ {t_ {i -1}}，i-1，i = 1，i = 1，a wortion a a wortion a a wortion a a wortion a a wortion a tht a worugion a a worugion a a worugion a a wortif函数，确保$ \hatθ_n$和$ \checkθ_n$坚固一致，并且序列$ \ sqrt {t_n}（\hatθ_n -θ）$和$ \ sqrt {t_n}
In this paper, we consider the statistical inference of the drift parameter $θ$ of non-ergodic Ornstein-Uhlenbeck~(O-U) process driven by a general Gaussian process $(G_t)_{t\ge 0}$. When $H \in (0, \frac 12) \cup (\frac 12,1) $ the second order mixed partial derivative of $R (t, s) = E [G_t G_s] $ can be decomposed into two parts, one of which coincides with that of fractional Brownian motion (fBm), and the other of which is bounded by $|ts|^{H-1}$. This condition covers a large number of common Gaussian processes such as fBm, sub-fractional Brownian motion and bi-fractional Brownian motion. Under this condition, we verify that $(G_t)_{t\ge 0}$ satisfies the four assumptions in references \cite{El2016}, that is, noise has Hölder continuous path; the variance of noise is bounded by the power function; the asymptotic variance of the solution $X_T$ in the case of ergodic O-U process $X$ exists and strictly positive as $T \to \infty$; for fixed $s \in [0,T)$, the noise $G_s$ is asymptotically independent of the ergodic solution $X_T$ as $T \to \infty$, thus ensure the strong consistency and the asymptotic distribution of the estimator $\tildeθ_T$ based on continuous observations of $X$. Verify that $(G_t)_{t\ge 0}$ satisfies the assumption in references \cite{Es-Sebaiy2019}, that is, the variance of the increment process $\{ ζ_{t_i}-ζ_{t_{i -1}}, i =1,..., n \}$ is bounded by the product of a power function and a negative exponential function, which ensure that $\hatθ_n$ and $\checkθ_n $ are strong consistent and the sequences $\sqrt{T_n} (\hat θ_n - θ)$ and $\sqrt {T_n} (\check θ_n - θ)$ are tight based on discrete observations of $X$
